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Relaxed locally correctable codes with nearly-linear block length and constant query complexity

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Chiesa, Alessandro, Gur, Tom and Shinkar, Igor (2021) Relaxed locally correctable codes with nearly-linear block length and constant query complexity. SIAM Journal of Computing . pp. 1395-1411. doi:10.1137/1.9781611975994.84 ISSN 0097-5397.

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Official URL: https://doi.org/10.1137/1.9781611975994.84

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Abstract

Locally correctable codes (LCCs) are codes C: Σk → Σn which admit local algorithms that can correct any individual symbol of a corrupted codeword via a minuscule number of queries. One of the central problems in algorithmic coding theory is to construct O(1)-query LCC with minimal block length. Alas, state-of-the-art of such codes requires exponential block length to admit O(1)-query algorithms for local correction, despite much attention during the last two decades.

This lack of progress prompted the study of relaxed LCCs, which allow the correction algorithm to abort (but not err) on small fraction of the locations. This relaxation turned out to allow constant-query correction algorithms for codes with polynomial block length. Specifically, prior work showed that there exist O(1)-query relaxed LCCs that achieve nearly-quartic block length n = k4+α, for an arbitrarily small constant α > 0.

We construct an O(1)-query relaxed LCC with nearly-linear block length n = k1+α, for an arbitrarily small constant α > 0. This significantly narrows the gap between the lower bound which states that there are no O(1)-query relaxed LCCs with block length n = k1+o(1). In particular, this resolves an open problem raised by Gur, Ramnarayan, and Rothblum (ITCS 2018).

Item Type: Journal Article
Subjects: Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Divisions: Faculty of Science, Engineering and Medicine > Science > Computer Science
Library of Congress Subject Headings (LCSH): Coding theory, Computer algorithms, Computational complexity
Journal or Publication Title: SIAM Journal of Computing
Publisher: SIAM
ISBN: 978-1-61197-599-4
ISSN: 0097-5397
Official Date: 2021
Dates:
DateEvent
2021Published
26 May 2021Accepted
Page Range: pp. 1395-1411
DOI: 10.1137/1.9781611975994.84
Status: Peer Reviewed
Publication Status: Published
Reuse Statement (publisher, data, author rights): “First Published in SIAM Journal of Computing in [volume and number, or year], published by the Society for Industrial and Applied Mathematics (SIAM)” and the copyright notice as stated in the article itself (e.g., “Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.”)
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 11 June 2021
Date of first compliant Open Access: 14 June 2021
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
MR/S031545/1UK Research and Innovationhttp://dx.doi.org/10.13039/100014013
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